Sensor signal prediction at unreported time periods

ABSTRACT

A method, a computer program product, and a system for predicting low-frequency sensor signal predictions using a hierarchical prediction model. The method includes receiving a historical dataset of high-frequency sensor signal data and low-frequency sensor signal data. The method also includes generating a Gaussian process regression model using the historical dataset and sensor parameters and outputting high-frequency sensor signal predictions. The method also includes generating a hierarchical Gaussian process model using the historical dataset and the high-frequency sensor signal predictions and predicting low-frequency sensor signal predictions.

BACKGROUND

The present disclosure relates to sensor signal processing, and morespecifically, to predicting sensor signals unreported from a sensortransmitting signal readings at a certain time period.

The Internet of Things (IoT) is a term commonly used to refer to asystem of interrelated computing devices, mechanical devices, digitaldevices, and sensors having the ability to transfer data over a network.IoT systems can be applied for consumer applications (e.g., smart home,elder care), organizational applications (e.g., healthcare,transportation, automation), and industrial applications (e.g.,manufacturing, agriculture).

Predictive modeling is a process that uses data and statistics topredict outcomes with data models. It is a mathematical approach thatuses an equation-based model to describe a phenomenon underconsideration. The data model can be used to forecast an outcome to somefuture state or time based upon changes to the model inputs.Additionally, the model parameters can explain how model inputsinfluence the outcome.

SUMMARY

Embodiments of the present disclosure include a computer-implementedmethod for predicting unmonitored sensor signals using a predictionmodel. The computer-implemented method includes receiving a historicaldataset of sensor signal data relating to an environment of a sensormonitoring system. The historical dataset includes a first sensor signaldata, a second sensor signal data, and input variables relating to thesensor monitoring system. The computer-implemented method also includesgenerating sensor signal responses relating to the first sensor signaldata by applying a Gaussian process regression model to the historicaldataset and sensor parameters. The computer-implemented method furtherincludes generating a hierarchical Gaussian process model that jointlyconsiders multi-dimensional covariance structures among the inputvariables, the first sensor signal data, and the second sensor signaldata, and predicting, by the hierarchical Gaussian process model, signalvalues relating to the second sensor signal data at time periods wherethe second sensor signal data was not monitored at using the sensorsignal responses.

Additional embodiments of the present disclosure include a computerprogram product for predicting low-frequency sensor signal predictionsusing a prediction model which can include computer-readable storagemedium having program instructions embodied therewith, the programinstruction executable by a processor to cause the processor to performa method. The method includes receiving a historical dataset of sensorsignal data relating to an environment of a sensor monitoring system.The historical dataset includes a first sensor signal data, a secondsensor signal data, and input variables relating to the sensormonitoring system. The method also includes generating sensor signalresponses relating to the first sensor signal data by applying aGaussian process regression model to the historical dataset and sensorparameters. The method further includes generating a hierarchicalGaussian process model that jointly considers multi-dimensionalcovariance structures among the input variables, the first sensor signaldata, and the second sensor signal data, and predicting, by thehierarchical Gaussian process model, signal values relating to thesecond sensor signal data at time periods where the second sensor signaldata was not monitored at using the sensor signal responses.

Further embodiments are directed to a system for predicting unmonitoredsensor signal using a prediction model and configured to perform themethod described above. The present summary is not intended toillustrate each aspect of every implementation of, and/or everyembodiment of the present disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the embodiments ofthe disclosure will become better understood with regard to thefollowing description, appended claims, and accompanying drawings where:

FIG. 1 is a block diagram illustrating an IoT system architecture, inaccordance with embodiments of the present disclosure.

FIG. 2 is a block diagram illustrating a IoT system architecture withtwo signal monitoring frequencies, in accordance with embodiments of thepresent disclosure.

FIG. 3 is a block diagram illustrating a data prediction system, inaccordance with embodiments of the present disclosure.

FIG. 4 is a hierarchical data structure of sensor signal data, inaccordance with embodiments of the present disclosure.

FIG. 5 is a flow diagram illustrating a process of predictinglow-frequency sensor signal data, in accordance with embodiments of thepresent disclosure.

FIG. 6 is a high-level block diagram illustrating an example computersystem that may be used in implementing one or more of the methods,tools, and modules, and any related functions, described herein, inaccordance with embodiments of the present disclosure.

FIG. 7 depicts a cloud computing environment, in accordance withembodiments of the present disclosure.

FIG. 8 depicts abstraction model layers, in accordance with embodimentsof the present disclosure.

While the present disclosure is amenable to various modifications andalternative forms, specifics thereof have been shown by way of example,in the drawings and will be described in detail. It should beunderstood, however, that the intention is not to limit the particularembodiments described. On the contrary, the intention is to cover allmodifications, equivalents, and alternatives falling within the scope ofthe present disclosure. Like reference numerals are used to designatelike parts in the accompanying drawings.

DETAILED DESCRIPTION

The present disclosure relates to sensor signal processing, and morespecifically, to predicting sensor signals unreported from a sensortransmitting signal readings at a certain time period. While the presentdisclosure is not necessarily limited to such applications, variousaspects of the disclosure may be appreciated through a discussion ofvarious examples using this context.

Systems, such as IoT systems, can collect and analyze data gathered fromsensors placed within the system. Sensor monitoring systems can usesensors in a wide array of applications. These applications include, forexample, environmental sensing, condition monitoring, and processautomation. Environmental sensing applications include sensors thatmonitor conditions such as air pollution, water pollution, productionmaterial pollution, and the like. Additionally, in hazardousenvironments, sensors can monitor natural disasters such as fire, flood,earthquakes, or landslides.

Process automation applications include sensor monitoring that providesusers with information regarding resources in a production process,supply chain status, material creation anomalies, performancemonitoring, process evaluation, as well improvements that can beimplemented into a process.

Limitations in sensor monitoring systems remain, however, as currentimplementations are limited by technological challenges. The deploymentand setup of a sensor monitoring system can be challenging. Forinstance, the environments where a sensor monitoring system is deployedto monitor environmental or production processes can be dynamic and candepend on the specific product, the phase of life of the product, or thekind of service provision considered. The type of product, or phase oflife in a production process, can have different requirements imposingdifferent constraints on a sensor monitoring system.

Additionally, a sensor monitoring system may limit the amount ofinformation it gathers over a period of time for various reasons. Thesereasons include, for example, the importance of a particular reading,computing resource restraints, the expense of reporting a metric,storage capacity, and the like. As such, sensors can be set to reportreadings at a lower or higher frequency, depending on the informationbeing gathered. As a result, some sensor data may be missing toaccurately perform analytics on the information gathered by a sensormonitoring system.

From a mathematical perspective, missing data can be divided into threetypes. The first type is called missing completely at random (MCAR),where the missingness does not depend on the variable of interest, orany other variable, which is observed in the dataset. The secondmechanism is missing at random (MAR), where there are observed valuesand missing input variables. The probability of missing data can bedependent on the observed input variables. The third mechanism ismissing not at random (MNAR), which dictates that the missingnessdepends on both observed and missing variables.

IoT modeling and sensor data prediction using different time-frequencydata can be considered as missing due to abnormal system operations(MCAR) and MNAR (missing due to limited computing resources,destructive, and expensive data collection process) mechanisms. A way toobtain an unbiased estimate of the parameters in an MNAR situation orMCAR situation is to model the missing data using predictive modeling.The model may then be incorporated into a more complex model forestimating missing values.

One approach to handling missing data is to use a data analysis method.An analysis method can be considered robust when there is little to nobias or distortion in the conclusions drawn on the population of data.However, this approach may not always be feasible. Alternatively,techniques such as data deletion and data imputation have also been usedin missing data situations. Deletion techniques exclude cases withmissing data and simply rely on complete data. This approach can removea sizable portion of the collected data and can diminish theeffectiveness of statistical analysis. Data imputation fill in predictedvalues at the locations of incomplete observations. Imputation canaccomplish this by using a mean/median/last-observation valuesubstitution, regression imputation, maximum likelihood method, multipleimputations, and the like. However, a major drawback of these methods isthe lack of uncertainty quantification associated with the prediction.

Embodiments of the present disclosure may overcome the above and otherproblems by using a data prediction system to accurately predictlow-frequency sensor data and to adjust sensor monitoring systemsconfigurations. The data prediction system can implement a hierarchicalGaussian process model approach to impute missing values inhigh-dimensional sensor data. The hierarchical Gaussian process modelcan be non-parametric with a new multi-dimensional covariance structureamong inputs and different time-frequency outputs that can imputemissing data in sensor form.

More specifically, the data prediction system can generate ahierarchical prediction model by inputting controllable sensorparameters into a Gaussian process regression model and output sensorpredictions for sensors that are monitored at a high-frequency. Thesensor predictions being outputted by the Gaussian process regressionmodel can be inputted into a hierarchical Gaussian Prediction model tooutput sensor predictions for sensors that are monitored at alow-frequency. Through the hierarchical modeling structure, thecorrelation between sensor parameters and high-frequency predictiondata, the correlation between sensor parameters and low-frequencyprediction data, and the correlation between high-frequency andlow-frequency prediction data are all considered to make a predictedsensor data for the low-frequency monitored sensor data.

Embodiments of the present disclosure include a feedback control forchanging sensor monitoring frequency in a sensor monitoring system. Thecollected sensor data in a sensor monitoring system may either beunnecessary because the monitoring frequency is too high, or it may beinadequate because the monitoring frequency is too low. The need tochange monitoring frequency may be due to a change (e.g., environmentalrelocation, new materials, etc.) in a sensor monitoring system orbecause some sensor readings may change in importance, requiring achange in frequency readings. The data prediction system can include afeedback control based on a predetermined error threshold compared withpredicted sensor data outputted by the data prediction system. Based onthe comparison, the data prediction system can recommend sensormonitoring frequencies that can improve the reliability of the predicteddata.

Embodiments of the present disclosure include a framework for updatingthe prediction model used by a sensor monitoring system based on achange in application. The data prediction system can generate asurrogate model for the sensor monitoring system. A multivariateGaussian process surrogate can be trained using the monitored,historical, and prediction data for a current application. Additionally,an uncertainty metric can be quantified based on the new application inwhich the sensor monitoring system is being applied. Using the surrogatemodel, unspecified parameters of the new application can be connectedwhile allowing for the model discrepancy and measurement errors.

By way of example, but not limitation, consider a sensor monitoringsystem configured as an IoT system monitoring the air quality of anindoor environment. The sensor monitoring system uses sensors to monitorthe temperature, humidity, and particle dust with 6,000 readings foreach sensor over a predetermined period (e.g., twelve hours, one day,one week). The frequency of the readings can be considered ashigh-frequency monitoring. Additionally, sensors also monitor carbonmonoxide and sulfur dioxide with 500 readings for each sensor over thesame predetermined period as the other sensors. The frequency of thesesensors can be considered low-frequency monitoring. 80% of the readingsfrom both the high-frequency data and low-frequency data can be used tobuild a hierarchical predictive model. The remaining 20% can be used totest the prediction error of the model.

Continuing with the example described immediately above, unspecifiedparameters, such as wind speed, wind direction, rainfall amount, andsolar radiation, can be combined with the data readings of thehigh-frequency monitoring to train a surrogate model. Data points froman outdoor environment can be introduced to further train and adapt thesurrogate model for use in an outdoor environment. The data predictionsystem, therefore, provides a framework for the sensor monitoring systemto allow it to transition from an indoor environment to an outdoorenvironment while also being able to accurately predict low-frequencydata.

Referring now to FIG. 1, shown is a high-level block diagram of anexemplary IoT platform 100 on which embodiments of the disclosure may beimplemented. The IoT platform includes IoT devices 110-1, 110-2, 110-3,110-4, 110-5 (collectively “IoT devices 110”), an IoT hub 120, a sensormonitoring system 125, a cellular network 130, a local network 135, theInternet 140, an IoT service 150, a website 160, and a user device 170.The IoT devices 110 are communicatively coupled over communicationchannels to the IoT hub 120. The IoT hub 120 is communicatively coupledto the IoT service 150 over the Internet 140.

The IoT devices 110 are components of the IoT platform configured withvarious types of sensors to collect information about themselves andtheir surroundings and provide the collected information to the IoTservice 150, to websites 160, and/or the user device 170. For example,the IoT devices 110 can be sensors such as temperature sensors,accelerometers, heat sensors, motion detectors, and the like. In someembodiments, the IoT devices 110 perform a specified function inresponse to a control command sent through the IoT hub 120. Asillustrated, the database 155 can store the data collected by the IoTdevices 110. The data stored in the database 155 can then be madeaccessible to an end-user via a computing device accessing the websites160 or through the user device 170.

The IoT hub 120 is a component of the IoT platform configured tomaintain and direct the IoT devices 110. The IoT hub 120 can establish aconnection to the Internet 140 via a cellular network 130 such as a 4G(e.g., Mobile WiMAX, LTE) or 5G cellular data service. Additionally, oralternatively, the IoT hub 120 can establish a connection to theInternet 140 using the local network 135 which can establish a Wi-Ficonnection through a Wi-Fi access point or router which couples the IoThub 120 to the Internet 140 (e.g., via an Internet Service Providerproviding Internet service to an end-user operating the IoT hub 120).

In some embodiments, the IoT platform 100 includes an IoT application orWeb application executable on the user device 170 to allow users toaccess and configure the IoT devices 110 (e.g., change monitoringfrequency of a sensor), the IoT hub 120, and/or the IoT service 150.

As illustrated, the sensor monitoring system 125 represents a collectionof IoT devices 110 connected to the IoT hub 120 installed in a singleenvironment (e.g., a home, business, warehouse) It should be noted, anynumber of IoT hubs 120 and IoT devices 110 can be installed in anenvironment to comprise the sensor monitoring system 125. Depending onthe application in which the sensor monitoring system 125 is deployedfor, additional or fewer IoT hubs 120 and IoT devices 110 may beinstalled and used.

It is noted that FIG. 1 is intended to depict the representative majorcomponents of an exemplary IoT platform 100. In some embodiments,however, individual components may have greater or lesser complexitythan as represented in FIG. 1, components other than or in addition toas shown in FIG. 1 may be present, and the number, type, andconfiguration of such components may vary.

FIG. 2 is a high-level block diagram of an exemplary IoT systemarchitecture 200. The exemplary IoT system architecture 200 illustratessensor information considered quantities of interest. Embodiments of thepresent disclosure can utilize these quantities of interest to predictmissing data. The IoT system architecture system inputs 210, an IoTsystem 220 (which can be the same as, or substantially similar to, IoTplatform 100 of FIG. 1), high-frequency sensor data 230, monitoringfrequency 235, low-frequency sensor data 240, and monitoring frequency245.

Initially, the system inputs 210 provide information to the IoT system220. The system inputs 210 include information provided to the IoTsystem 220 that allow the system 220 to operate within an environmentsuch as sensor settings, configurations, placement, and the like.

The exemplary IoT system 220 outputs high-frequency sensor data 230 andlow-frequency sensor data 240 using the system inputs 210 and sensorsreporting to the IoT system 220. In some circumstance, restrictions mayexist preventing the IoT system 200 to receive and/or monitor thelow-frequency sensor data 240 at a higher frequency.

Embodiments of the present disclosure can use the correlation betweenthe monitoring frequency 235 of the high-frequency sensor data 230 withthe monitoring frequency 245 of the low-frequency sensor data to imputemissing values in the low-frequency sensor data 240. These missingvalues may provide useful insights in the operation of the IoT system220.

FIG. 3 is a block diagram illustrating a data prediction system 300 forpredicting low-frequency sensor signals, in accordance with embodimentsof the present disclosure. The data prediction system 300 includes asensor database 310, a Gaussian process regression model 320, ahierarchical Gaussian process model 330, a feedback control 340, ahistorical database 350, and a model updating framework 360.

The sensor database 310 is a database in the data prediction system 300configured to store configured to store a historical dataset of sensorsignal data from sensors in a sensor monitoring system. The sensorsignal data can include sensor signal data collected from sensorsreporting at different time frequencies. For example, a sensor signaldata may be collected at a low-frequency (e.g., ten minutes, one hour,one day) while another sensor signal data may be collected at a hightime-frequency (e.g., every second, thirty seconds, every minute). Itshould be noted that low-frequency sensor signal data and high-frequencysensor signal data can be relative to each other. High-frequency sensorsignal data need only be collected more often than low-frequency sensorsignal data. It should be noted that while the disclosure discusses anevaluation of sensor signals at higher and lower frequencies,embodiments of the present disclosure can also use sensor signals atvarious time-frequencies. For example, the sensor signal data caninclude sensor signals of the same frequency but at different reportingtimes. While shown in FIG. 3 as a database, the sensor database 310 canbe a mapping, a table, journal, metadata, and the like.

In some embodiments, the sensor database 310 stored the historicaldataset of sensor signal data remotely. For example, the sensor database310 can be the same as, or substantially similar to, database 155 ofFIG. 1. The sensor database 310 can be accessed by the data predictionsystem 300 over the Internet via a computing device.

The Gaussian process regression model 320 is a component of the dataprediction system 300 configured to predict high-frequency sensor signaldata. The Gaussian process regression model 320 can model ahigh-frequency prediction variable as a noise Gaussian processregression model, as illustrated by Equation 1 described below:

y _(i) ^(H)˜

(ƒ(x _(i)),σ₁ ²   Equation 1

where y_(i) ^(H) represents a high-frequency prediction variable as anoise version Gaussian process regression model of the latent functionvalue ƒ(x_(i)). The distribution of noise can be Gaussian N(0,σ₁ ²) withzero mean and variance 61. The model can be additionally defined byEquation 2 described below:

ƒ(x)˜

(μ,K)   Equation 2

where a zero mean (μ=0) and a kernel function K (which is the covariancematrix over observations with a hyper-parameter θ) as a covariancematrix.

From the above definitions, the Gaussian process regression model 320can get a joint probability of the prediction variables and latentfunction variable p(y^(H),ƒ)=p(y^(H)|ƒ)p(ƒ). The distribution of thelatent function value ƒ is a Gaussian distribution with mean andcovariance illustrated by Equation 3 and 4 described below:

mean(ƒ(x*)|x,y ^(H))=k _(X)(σ₁ ² I+K _(xx))⁻¹ y ^(H)   Equation 3

Cov(ƒ(x*)|x,y ^(H))=k _(x*x*) −k _(x*X)(σ₁ ² I+K _(xx))⁻¹ k _(Xx*)  Equation 4

where k_(x*X)=k(x*,X) is an n-dimensional row vector of the covariancebetween x* and N training samples, K_(xx)=k(X,X) denotes the kernelfunction of the N training samples (y^(H),X), which can be used toestimate the covariance function. The related predictive distribution ofy^(H)* can also be Gaussian with the mean defined by mean(ƒ(x*)|x,y^(H)) and covariance defined by Cov(ƒ(x*)|x,y^(H))+σ₁ ²I.

The Gaussian process regression model 320 can also factor inhyperparameters for prediction. The hyperparameters can be controllablesensor parameters included in the historical dataset that is stored inthe sensor database 310. Using the definitions described above, thehyperparameters can be indicated using θ. In a non-Bayesian analysis,the mean E(ƒ(x*)|D), can be evaluated at the maximum likelihoodestimation (MLE) of θ by taking the log-likelihood function, which canbe used as a prediction for y^(H)*. In a Bayesian approach, priorinformation about the hyperparameter can be summarized in the form of aprior density indicated by p(θ). The posterior density for θ given atraining dataset Q can be defined as p (θ|Q)˜p(θ)p(y|X,θ); wherep(y|X,θ) is the density function of an N-dimensional multivariate normaldistribution with zero mean and covariance matrix (σ²I+K_(xx)).

The hierarchical Gaussian process model 330 is a component of the dataprediction system 300 configured to predict sensor data at ahigh-frequency given the outputted predicted high-frequency sensorsignal data produced by the Gaussian process regression model 320. Thehierarchical Gaussian process model 330 can leverage the relationshipbetween high-frequency sensor signal data and low-frequency sensorsignal data using the input received by the Gaussian process regressionmodel 320 as illustrated by equations 5 and 6 described below:

Y ^(L) _(i)˜

(ƒ(y ^(H) _(i)),σ₂ ²)   Equation 5

ƒ(y ^(H) _(i))˜

(μ,C)   Equation 6

where y^(H) represents high-frequency sensor signal data, y^(L)represents low-frequency sensor signal data, y_(i) ^(H) represents inputreceived from the Gaussian process regression model 320. Additionally,y^(L) can represent a continuous longitudinal variable that follows aGaussian distribution in equation 5, where ƒ(y_(i) ^(H)) follows theGaussian process distribution with the mean μ is assumed as zero (whichis separate from the mean of the predicted y^(H)). The covarianceC=Cov(y^(H),y^(L)) can be estimated using a Markov Chain Monte Carloapproach (MCMC). The low-frequency sensor signal data can then bepredicted using Equation 7 described below:

E(y ^(L) *|D)=c _(y) H* _(y) H*(σ₂ ² +C)⁻¹ y ^(L)   Equation 7

where C_(y)H*_(y)H*=Cov(ƒ(x*)|x,y^(H))+σ₁ ²I can be calculated asdescribed above, and C=Cov(y^(H),y^(L)) can be estimated by MCMC.

The feedback control 340 is a component of the data prediction system300 configured to provide frequency feedback for the sensors in a sensormonitoring system. Default frequency settings for sensors may notaccurately monitor an environment. High-frequency sensor signal data mayultimately be meaningless and considered redundant. While some sensorsignal data may be inadequately monitored. The feedback control 340 canprovide feedback for sensor frequency by performing cross-validation ofthe high-frequency sensor signal data with the low-frequency sensorsignal data to calculate prediction errors.

The feedback control 340 is further configured to plot the predictionerrors for the different sensor parameters of the given sensors. For anysensor signal, the area where the absolute percentage error of onesensor signal is larger than a user-defined threshold can trigger theneed for that particular sensor to be monitored at a higher frequency.Otherwise, the monitoring frequency for that sensor can be lowered.

In some embodiments, the feedback control 340 operates as an automaticcontrol loop that dynamically changes the sensor monitoring frequenciesof sensors in a sensor monitoring system. The feedback control 340 cancontinuously receive low-frequency sensor signal data predictions fromthe hierarchical Gaussian process model 330 and validates the data byplotting the prediction errors and determining whether a reading exceedsthe pre-defined user threshold. The control loop can continue until theprediction errors reach a pre-defined acceptable level.

The historical database 350 is a database in the data prediction system300 configured to store a historical dataset of sensor signal data fromsensors in a sensor monitoring system used in a different application.For example, the sensor database 310 can store sensor signal data of asensor monitoring system installed in an indoor environment, and thehistorical database 350 can store sensor signal data of the sensormonitoring system installed in an outdoor environment. The sensor signaldata can include sensor signal data collected from sensors reporting atdifferent time frequencies. For example, a sensor signal data may becollected at a low-frequency (e.g., ten minutes, one hour, one day)while another sensor signal data may be collected at a hightime-frequency (e.g., every second, thirty seconds, every minute). Itshould be noted that low-frequency sensor signal data and high-frequencysensor signal data can be relative to each other. High-frequency sensorsignal data need only be collected more often than low-frequency sensorsignal data. While shown in FIG. 3 as a database, the historicaldatabase 350 can be a mapping, a table, journal, metadata, and the like.

In some embodiments, the historical database 350 stores the historicaldataset of sensor signal data remotely. For example, the historicaldatabase 350 can be the same as, or substantially similar to, database155 of FIG. 1. The historical database 350 can be accessed by the dataprediction system 300 over the Internet via a computing device.

The model updating framework 360 is a component of the data predictionsystem 300 configured to update a prediction model of a sensormonitoring system applied to different applications. The model updatingframework 360 is configured to generate a surrogate prediction model fora sensor monitoring system that is already modeled by the dataprediction system 300. The surrogate prediction model can be generatedby training a multivariate Gaussian process surrogate model using thesensor data stored in the sensor database 310 and the sensor data storedin the historical database 350. Additionally, controllable sensor inputparameters can also be inputted.

Unspecified parameters relating to a specific application can beadjusted to adapt the sensors to different system applications. Forexample, a sensor is monitoring the formation of a material using aspecific type of metal (e.g., iron) can be considered one type ofapplication. Another application can be the same sensor monitoring theformation of the material using a different type of metal (e.g.,cobalt). The unspecified parameters can be used to capture the variancebetween applications so as to allow the sensor to accurately monitor theenvironment.

The model updating framework 360 is further configured to quantify anuncertainty of changing the application of a sensor monitoring system.The model updating framework 360 can connect the unspecified parametersto the monitored sensor data under a different application and use thesurrogate prediction model. In some embodiments, the model updatingframework 360 quantifies an uncertainty of changing the application of asensor monitoring system using equation 8 as described below:

Z(x)=Ŷ(x,θ)+δ(x)+ε   Equation 8

where x represents input parameters for the controllable sensorsettings, θ here represents the unspecified parameters adjustable toadapt a sensor to different applications, Z represents a differentapplication using the built surrogate model (Ŷ(x,θ)) of sensor signaldata. The model discrepancy can be represented by (δ(x)) withmeasurement errors (ε).

Continuing with equation 8, prior probabilities for all unknownparameters represented by Ø, the unspecified parameters θ, andhyper-parameters (e.g., the covariance of the discrepancy function Σδcan be decided on. Additionally, the model updating framework 360 canassign a function that gets Ø as input and computes the priorprobability for that high-dimensional variable space. The model updatingframework 360 can also compute a function that gets Ø and the monitoredsensor parameters to compute a posterior probability for maximizinglog-likelihood and MCMC sampling. Using a single componentMetropolis-Hastings sampling, the model updating framework 360 can findposteriors for each unknown parameter. The unspecified parameters canthen be specified using the posterior distributions.

In some embodiments, the model updating framework 360 uses a Krigingtechnique to make final predictions. Using the unspecified parametersspecified by posterior distributions, the Kriging technique can be togenerate final predictions as represented by equation 9 described below:

Y ^(P) =E[Y ^(P) |Y ^(E)]=μ(X ^(P))+Cov^(PE)(Cov^(EE))⁻¹(Y ^(E)−μ(X^(E)))   Equation 9

where Y^(P) represents the final predicted outputs for new inputs X^(P).For computing the outputs Y^(P), the model updating framework 360 canuse the monitored sensor parameters X^(Z) in addition to running asimulation of the prediction model with the unspecified parametersspecified as posterior distributions.

It is noted that FIG. 3 is intended to depict the representative majorcomponents of an exemplary data prediction system 300. In someembodiments, however, individual components may have greater or lessercomplexity than as represented in FIG. 3, components other than or inaddition to as shown in FIG. 3 may be present, and the number, type, andconfiguration of such components may vary.

FIG. 4 is an exemplary hierarchical structure 400 of sensor signal datain descending order, in accordance with embodiments of the presentdisclosure. The hierarchical structure 400 illustrates amulti-dimensional hierarchical covariance structure among inputvariables, high-frequency outputs, and low-frequency outputs in ahierarchical way. Its posterior prediction is to impute missing sensordata in a sensor monitoring system.

Inputs 410 represent new inputs X* into the data prediction system 400.The data prediction system 400 can hierarchically predict the low tohigh-frequency response of the new inputs 410. To do so, the dataprediction system 400 generates the hierarchical structure 400 bycomputing two output levels 420, 430. The first level 420 is acovariance among the inputs 410 and the highest frequency sensor signaldata output.

The second level 430 represents the covariance of the highesttime-frequency sensor signal data and the second-highest time-frequencysensor signal data. A correlation between the two outputs can be modeled440, 450, with the amount of data in the low-frequency sensor data. Themodels 440, 450 can keep adding the covariance levels until the last twolowest time-frequency outputs.

FIG. 5 is a flow diagram illustrating a process 500 of predictinglow-frequency sensor signal data. The process 500 begins by receiving ahistorical dataset of sensor signal data of a sensor monitoring system.The historical dataset includes a firsts sensor signal data, a secondsensor signal data, and input variables relating to an environment ofthe sensor monitoring system. For example, the environment can be asensor monitoring system monitoring environmental conditions in anindoor environment. Another environment can be the same sensormonitoring system monitoring environmental conditions in an outdoorenvironment. In some embodiments, the first sensor signal data includesreadings at a higher rate than the second sensor signal data. Forexample, the first sensor signal data can include sensor reading everyfifteen seconds while the second sensor signal data can include sensorreadings every minute. The signal data in the historical dataset cancorrespond to sensors positioned within the sensor monitoring system.

In some embodiments, the historical dataset is received remotely from asensor database 310 collecting and storing sensor data of the sensormonitoring system. For example, the sensor monitoring system can be anIoT system that collects and stores sensor data over a network where thesensor database 310 is accessible via a computing device.

Sensor signal responses are generated using a Gaussian regressionprocess through the use of a Gaussian regression model 320. This isillustrated at step 520. The Gaussian process regression model 320 takesas input, the input variables from the historical dataset. The inputvariables can be sensor parameters of the sensors used in the sensormonitoring system. Additionally, the first sensor signal data is alsoinputted into the Gaussian process regression model. The input variablescan also include pre-defined sensor parameters that are set based on anenvironment. These pre-defined parameters can assist the sensors inaccurately reporting readings. The parameters can also vary based on thesensor type, as well as the environment in which the sensor isinstalled.

The Gaussian process regression model 320 can model a high-frequencysensor signal prediction as a noise-version of a latent function valueusing the sensor parameters and the first sensor signal data. In someembodiments, the latent function value is defined by equation 1, definedabove, where the distribution of noise is Gaussian represented by a zeromean and variance. Once modeled, the Gaussian process regression model320 outputs the sensor signal responses.

A hierarchical Gaussian process model 330 is generated using ahierarchical Gaussian regression process. This is illustrated at step530. The hierarchical Gaussian process model 330 can be fitted to thefirst sensor signal data and the second sensor signal data located inthe historical dataset in order to leverage their correlatedrelationship while using the sensor signal responses produced by theGaussian process regression model 320.

The hierarchical Gaussian process model is non-parametric with a newmulti-dimensional covariance structure among inputs and different timefrequency outputs which offers a greater level of flexibility to imputemissing data in the sensor form. To portray the inherent hierarchicalstructure, the signal data is sorted in descending order based on theirmonitoring frequencies. Additionally, the hierarchical Gaussian processmodel considers the multi-dimensional covariance in different levels.

The covariance among the first sensor signal data and the second signaldata represents the first level. The first level covariance can have thehighest influence on IoT missing data imputation, as these historicaldata tend to include the fullest information of the dependent inputs andoutputs.

The covariance of the highest time frequency output and the secondhighest time frequency output is in the second level. The correlationbetween two outputs with the amount of data in the lower time frequencydata can then be modeled. Additional levels are applied until the lasttwo lowest time frequency outputs are reached.

In some embodiments, the hierarchical Gaussian process model predictssensor signal data relating to the second sensor signal data based on acorrelation between the first sensor signal data and the second sensorsignal data. Additionally, the hierarchical Gaussian process modeljointly considers multi-dimensional covariance structures among theinput variables, the first sensor signal data, and the second sensorsignal data. A posterior prediction of the hierarchical Gaussian processmodel can also impute missing sensor data in the sensor monitoringsystem.

The hierarchical Gaussian process model 330 can model sensor signalvalues as a noise-version of the sensor signal responses. In someembodiments, the latent function value is defined by equation 1, definedabove, where the distribution of noise is Gaussian represented by a zeromean and variance.

The hierarchical Gaussian process model 330 predicts sensor signalvalues. This is illustrated at step 540. In some embodiments, thehierarchical Gaussian process model 330 predicts the sensor signalvalues using equation 7 defined above. The predictions can representlow-frequency sensor signals that are missing from the historicaldataset and are determined based on the correlation between thehigh-frequency sensor signals and the low-frequency sensor signals.

The data prediction system 300 builds a feedback control 340 providingthe sensor monitoring system with adjustments. This is illustrated atstep 550. The feedback control 340 can perform a cross-validation of thesensor signal responses with the signal values to determine a predictionerror that provides a feedback control to the sensor monitoring system.The sensors parameters can be the same sensor parameters used in makingthe sensor signal predictions or from all sensor parameters of sensorsin the sensor monitoring system.

The feedback control 340 can plot the prediction errors with all sensorparameters provided by the sensor monitoring system and determine theprediction errors exceeding a predetermined threshold. For example, ifthe prediction error exceed five percent then those predictions areselected as needing adjustment.

The feedback control 340 can adjust a reporting frequency for thesensors based on the prediction error. The adjustment can be to adjust asensor to a lower frequency level or to a high-frequency level dependingon the prediction error produced by the cross-validation. The feedbackcontrol 340 can continue to perform this process in a loop until theprediction error is within the predetermined threshold set by a user.

The model updating framework 360 updates the framework of the dataprediction system 300 to provide a framework for transitioning thesensor monitoring system to a different environment This is illustratedat step 560. The model updating framework 360 generates a surrogateprediction model for the sensor monitoring system. The surrogateprediction model can include a Gaussian process regression model and ahierarchical Gaussian process model configured to the differentenvironment.

The model updating framework 360 can train the surrogate predictionmodel using the historical dataset and a second historical dataset. Thesecond historical dataset can be stored in the historical database 350and includes other first sensor signal data and other second sensorsignal data for the sensor monitoring system relating to the differentenvironment.

The model updating framework 360 also quantifies an uncertainty relatingto the sensor monitoring system applied in the second application. Theuncertainty can be quantified using the surrogate prediction model andunspecified parameters of the sensors in the sensor monitoring system.Based on the uncertainty, the model updating framework 360 can computean update to the hierarchical Gaussian process model and the Gaussianprocess regression model applied to the second application. This allowsthe data prediction system 300 to accurately make signal values for thesensor monitoring system in the different environment.

Referring now to FIG. 6, shown is a high-level block diagram of anexample computer system 600 (e.g., the data prediction system 300) thatmay be used in implementing one or more of the methods, tools, andmodules, and any related functions, described herein (e.g., using one ormore processor circuits or computer processors of the computer), inaccordance with embodiments of the present disclosure. In someembodiments, the major components of the computer system 600 maycomprise one or more processors 602, a memory 604, a terminal interface612, an I/O (Input/Output) device interface 614, a storage interface616, and a network interface 618, all of which may be communicativelycoupled, directly or indirectly, for inter-component communication via amemory bus 603, an I/O bus 608, and an I/O bus interface 610.

The computer system 600 may contain one or more general-purposeprogrammable central processing units (CPUs) 602-1, 602-2, 602-3, and602-N, herein generically referred to as the processor 602. In someembodiments, the computer system 600 may contain multiple processorstypical of a relatively large system; however, in other embodiments, thecomputer system 600 may alternatively be a single CPU system. Eachprocessor 601 may execute instructions stored in the memory 604 and mayinclude one or more levels of on-board cache.

The memory 604 may include computer system readable media in the form ofvolatile memory, such as random-access memory (RAM) 622 or cache memory624. Computer system 600 may further include otherremovable/non-removable, volatile/non-volatile computer system storagemedia. By way of example only, storage system 626 can be provided forreading from and writing to a non-removable, non-volatile magneticmedia, such as a “hard drive.” Although not shown, a magnetic disk drivefor reading from and writing to a removable, non-volatile magnetic disk(e.g., a “floppy disk”), or an optical disk drive for reading from orwriting to a removable, non-volatile optical disc such as a CD-ROM,DVD-ROM or other optical media can be provided. In addition, the memory604 can include flash memory, e.g., a flash memory stick drive or aflash drive. Memory devices can be connected to memory bus 603 by one ormore data media interfaces. The memory 604 may include at least oneprogram product having a set (e.g., at least one) of program modulesthat are configured to carry out the functions of various embodiments.

Although the memory bus 603 is shown in FIG. 6 as a single bus structureproviding a direct communication path among the processors 602, thememory 604, and the I/O bus interface 610, the memory bus 603 may, insome embodiments, include multiple different buses or communicationpaths, which may be arranged in any of various forms, such aspoint-to-point links in hierarchical, star or web configurations,multiple hierarchical buses, parallel and redundant paths, or any otherappropriate type of configuration. Furthermore, while the I/O businterface 610 and the I/O bus 608 are shown as single respective units,the computer system 600 may, in some embodiments, contain multiple I/Obus interface units, multiple I/O buses, or both. Further, whilemultiple I/O interface units are shown, which separate the I/O bus 608from various communications paths running to the various I/O devices, inother embodiments some or all of the I/O devices may be connecteddirectly to one or more system I/O buses.

In some embodiments, the computer system 600 may be a multi-usermainframe computer system, a single-user system, or a server computer orsimilar device that has little or no direct user interface but receivesrequests from other computer systems (clients). Further, in someembodiments, the computer system 600 may be implemented as a desktopcomputer, portable computer, laptop or notebook computer, tabletcomputer, pocket computer, telephone, smartphone, network switches orrouters, or any other appropriate type of electronic device.

It is noted that FIG. 6 is intended to depict the major representativecomponents of an exemplary computer system 600. In some embodiments,however, individual components may have greater or lesser complexitythan as represented in FIG. 6, components other than or in addition tothose shown in FIG. 6 may be present, and the number, type, andconfiguration of such components may vary.

One or more programs/utilities 628, each having at least one set ofprogram modules 630 (e.g., the data prediction system 300), may bestored in memory 604. The programs/utilities 628 may include ahypervisor (also referred to as a virtual machine monitor), one or moreoperating systems, one or more application programs, other programmodules, and program data. Each of the operating systems, one or moreapplication programs, other program modules, and program data or somecombination thereof, may include an implementation of a networkingenvironment. Programs 628 and/or program modules 630 generally performthe functions or methodologies of various embodiments.

It is to be understood that although this disclosure includes a detaileddescription on cloud computing, implementation of the teachings recitedherein is not limited to a cloud computing environment. Rather,embodiments of the present invention are capable of being implemented inconjunction with any other type of computing environment now known orlater developed.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g., networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines, and services) that canbe rapidly provisioned and released with minimal management effort orinteraction with a provider of the service. This cloud model may includeat least five characteristics, at least three service models, and atleast four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded automatically without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneousthin or thick client platforms (e.g., mobile phones, laptops, andP.D.A.s).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or datacenter).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned, in some cases automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer, the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active user accounts). Resource usage can bemonitored, controlled, and reported, providing transparency for both theprovider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isto use the provider's applications running on a cloud infrastructure.The applications are accessible from various client devices through athin client interface such as a web browser (e.g., web-based e-mail).The consumer does not manage or control the underlying cloudinfrastructure including network, servers, operating systems, storage,or even individual application capabilities, with the possible exceptionof limited user-specific application configuration settings.

Platform as a Service (PaaS): the capability provided to the consumer isto deploy onto the cloud infrastructure consumer-created or acquiredapplications created using programming languages and tools supported bythe provider. The consumer does not manage or control the underlyingcloud infrastructure including networks, servers, operating systems, orstorage, but has control over the deployed applications and possiblyapplication hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is to provision processing, storage, networks, and otherfundamental computing resources where the consumer is able to deploy andrun arbitrary software, which can include operating systems andapplications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or off-premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting forload-balancing between clouds).

A cloud computing environment is service-oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure that includes anetwork of interconnected nodes.

Referring now to FIG. 7, illustrative cloud computing environment 700 isdepicted. As shown, cloud computing environment 700 includes one or morecloud computing nodes 710 with which local computing devices used bycloud consumers, such as, for example, personal digital assistant(P.D.A.) or cellular telephone 720-1, desktop computer 720-2, laptopcomputer 720-3, and/or automobile computer system 720-4 may communicate.Nodes 710 may communicate with one another. They may be grouped (notshown) physically or virtually, in one or more networks, such asPrivate, Community, Public, or Hybrid clouds as described hereinabove,or a combination thereof. This allows cloud computing environment 700 tooffer infrastructure, platforms and/or software as services for which acloud consumer does not need to maintain resources on a local computingdevice. It is understood that the types of computing devices 720-1 to720-4 shown in FIG. 7 are intended to be illustrative only and thatcomputing nodes 710 and cloud computing environment 700 can communicatewith any type of computerized device over any type of network and/ornetwork addressable connection (e.g., using a web browser).

Referring now to FIG. 8, a set of functional abstraction layers 800provided by cloud computing environment 700 (FIG. 7) is shown. It shouldbe understood in advance that the components, layers, and functionsshown in FIG. 8 are intended to be illustrative only and embodiments ofthe invention are not limited thereto. As depicted, the following layersand corresponding functions are provided:

Hardware and software layer 810 includes hardware and softwarecomponents. Examples of hardware components include mainframes 811; RISC(Reduced Instruction Set Computer) architecture-based servers 812;servers 813; blade servers 814; storage devices 815; and networks andnetworking components 816. In some embodiments, software componentsinclude network application server software 817 and database software818.

Virtualization layer 820 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers821; virtual storage 822; virtual networks 823, including virtualprivate networks; virtual applications and operating systems 824; andvirtual clients 825.

In one example, management layer 830 may provide the functions describedbelow. Resource provisioning 831 provides dynamic procurement ofcomputing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and Pricing 832provide cost tracking as resources are utilized within the cloudcomputing environment, and billing or invoicing for consumption of theseresources. In one example, these resources may include applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks, as well as protection for data and other resources.User portal 833 provides access to the cloud computing environment forconsumers and system administrators. Service level management 834provides cloud computing resource allocation and management such thatrequired service levels are met. Service Level Agreement (S.L.A.)planning and fulfillment 835 provide pre-arrangement for, andprocurement of, cloud computing resources for which a future requirementis anticipated in accordance with an S.L.A.

Workloads layer 840 provides examples of functionality for which thecloud computing environment may be utilized. Examples of workloads andfunctions which may be provided from this layer include mapping andnavigation 841; software development and lifecycle management 1342(e.g., the data prediction system 300); virtual classroom educationdelivery 843; data analytics processing 844; transaction processing 845;and precision cohort analytics 846.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer-readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer-readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer-readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer-readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (R.O.M.), an erasable programmable read-only memory(EPROM or Flash memory), a static random access memory (SRAM), aportable compact disc read-only memory (CD-ROM), a digital versatiledisk (DVD), a memory stick, a floppy disk, a mechanically encoded devicesuch as punch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer-readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer-readable program instructions described herein can bedownloaded to respective computing/processing devices from acomputer-readable storage medium or to an external computer or externalstorage device via a network, for example, the Internet, a local areanetwork, a wide area network and/or a wireless network. The network maycomprise copper transmission cables, optical transmission fibers,wireless transmission, routers, firewalls, switches, gateway computersand/or edge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (I.S.A.) instructions, machineinstructions, machine dependent instructions, microcode, firmwareinstructions, state-setting data, configuration data for integratedcircuitry, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as Smalltalk, C++, or the like, andprocedural programming languages, such as the “C” programming languageor similar programming languages. The computer readable programinstructions may execute entirely on the user's computer, partly on theuser's computer, as a standalone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer may beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection may be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (P.L.A.) may execute the computer readable program instructionsby utilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a computer, or other programmable data processing apparatusto produce a machine, such that the instructions, which execute via theprocessor of the computer or other programmable data processingapparatus, create means for implementing the functions/acts specified inthe flowchart and/or block diagram block or blocks. These computerreadable program instructions may also be stored in a computer readablestorage medium that can direct a computer, a programmable dataprocessing apparatus, and/or other devices to function in a particularmanner, such that the computer readable storage medium havinginstructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be accomplished as one step, executed concurrently,substantially concurrently, in a partially or wholly temporallyoverlapping manner, or the blocks may sometimes be executed in thereverse order, depending upon the functionality involved. It will alsobe noted that each block of the block diagrams and/or flowchartillustration, and combinations of blocks in the block diagrams and/orflowchart illustration, can be implemented by special purposehardware-based systems that perform the specified functions or acts orcarry out combinations of special purpose hardware and computerinstructions.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the variousembodiments. As used herein, the singular forms “a,” “an,” and “the” areintended to include the plural forms as well, unless the context clearlyindicates otherwise. It will be further understood that the terms“includes” and/or “including,” when used in this specification, specifythe presence of the stated features, integers, steps, operations,elements, and/or components, but do not preclude the presence oraddition of one or more other features, integers, steps, operations,elements, components, and/or groups thereof. In the previous detaileddescription of example embodiments of the various embodiments, referencewas made to the accompanying drawings (where like numbers represent likeelements), which form a part hereof, and in which is shown by way ofillustration specific example embodiments in which the variousembodiments may be practiced. These embodiments were described insufficient detail to enable those skilled in the art to practice theembodiments, but other embodiments may be used and logical, mechanical,electrical, and other changes may be made without departing from thescope of the various embodiments. In the previous description, numerousspecific details were set forth to provide a thorough understanding thevarious embodiments. But the various embodiments may be practicedwithout these specific details. In other instances, well-known circuits,structures, and techniques have not been shown in detail in order not toobscure embodiments.

When different reference numbers comprise a common number followed bydiffering letters (e.g., 100a, 100b, 100c) or punctuation followed bydiffering numbers (e.g., 100-1, 100-2, or 100.1, 100.2), use of thereference character only without the letter or following numbers (e.g.,100) may refer to the group of elements as a whole, any subset of thegroup, or an example specimen of the group.

Further, the phrase “at least one of,” when used with a list of items,means different combinations of one or more of the listed items can beused, and only one of each item in the list may be needed. In otherwords, “at least one of” means any combination of items and number ofitems may be used from the list, but not all of the items in the listare required. The item can be a particular object, a thing, or acategory.

For example, without limitation, “at least one of item A, item B, oritem C” may include item A, item A and item B, or item B. This examplealso may include item A, item B, and item C or item B and item C. Ofcourse, any combinations of these items can be present. In someillustrative examples, “at least one of” can be, for example, withoutlimitation, two of item A; one of item B; and ten of item C; four ofitem B and seven of item C; or other suitable combinations.

Different instances of the word “embodiment” as used within thisspecification do not necessarily refer to the same embodiment, but theymay. Any data and data structures illustrated or described herein areexamples only, and in other embodiments, different amounts of data,types of data, fields, numbers and types of fields, field names, numbersand types of rows, records, entries, or organizations of data may beused. In addition, any data may be combined with logic, so that aseparate data structure may not be necessary. The previous detaileddescription is, therefore, not to be taken in a limiting sense.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration but are not intended tobe exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

Although the present invention has been described in terms of specificembodiments, it is anticipated that alterations and modification thereofwill become apparent to the skilled in the art. Therefore, it isintended that the following claims be interpreted as covering all suchalterations and modifications as fall within the true spirit and scopeof the invention.

What is claimed is:
 1. A computer-implemented method of predictingunmonitored sensor signals using a prediction model, thecomputer-implemented method comprising: receiving a historical datasetof sensor signal data relating to an environment of a sensor monitoringsystem, wherein the historical dataset includes a first sensor signaldata, a second sensor signal data, and input variables relating to thesensor monitoring system; generating sensor signal responses relating tothe first sensor signal data by applying a Gaussian process regressionmodel to the historical dataset and sensor parameters; generating ahierarchical Gaussian process model that jointly considersmulti-dimensional covariance structures among the input variables, thefirst sensor signal data, and the second sensor signal data; andpredicting, by the hierarchical Gaussian process model, signal valuesrelating to the second sensor signal data at time periods where thesecond sensor signal data was not monitored at using the sensor signalresponses.
 2. The computer-implemented method of claim 1, furthercomprising: generating a surrogate prediction model for the sensormonitoring system to provide a framework for transitioning the sensormonitoring system to a different environment; training the surrogateprediction model using the historical dataset and a second historicaldataset, wherein the second historical dataset includes other firstsensor signal data and other second sensor signal data relating to asecond application; quantifying an uncertainty relating to the sensormonitoring system in the different environment using the surrogateprediction model and unspecified parameters of sensors in the sensormonitoring system relating to the different environment; computingparameter settings of unknown parameters relating to the differentenvironment by applying a kriging technique to known parameters of thedifferent environment; and providing the surrogate prediction model withthe computed parameter settings as the framework for the sensormonitoring system in the different environment.
 3. Thecomputer-implemented method of claim 1, further comprising: determiningprediction errors relating to the sensor parameters by performing across-validation of the sensor signal responses with the signal valuesto provide a feedback control to the sensor monitoring system; plottingthe prediction errors with all sensor parameters provided by the sensormonitoring system; determining the prediction errors exceeding apredetermined threshold; and adjusting a reporting frequency for sensorsin the sensor monitoring system based on the prediction errors exceedingthe predetermined threshold.
 4. The computer-implemented method of claim1, wherein generating the hierarchical Gaussian process model comprises:sorting the first sensor signal data and the second sensor signal datain a descending order based on their monitoring frequencies to portraytheir hierarchical structure; determining a covariance among the inputvariables and a highest time frequency signal data at a first level ofthe hierarchical structure; producing a second level of the hierarchicalstructure using the covariance of the highest time frequency sensorsignal data and the covariance of a second highest time frequency sensorsignal data; and applying additional levels to the hierarchicalstructure until covariances of two lowest time frequency sensor signaldata is added to a level.
 5. The computer-implemented method of claim 1,wherein the hierarchical Gaussian process model predicts sensor signaldata relating to the second sensor signal data based on a correlationbetween the first sensor signal data and the second sensor signal data.6. The computer-implemented method of claim 1, wherein the hierarchicalGaussian process model jointly considers multi-dimensional covariancestructures among the input variables, the first sensor signal data, andthe second sensor signal data.
 7. The computer-implemented method ofclaim 1, wherein a posterior prediction of the hierarchical Gaussianprocess model imputes missing sensor data in the sensor monitoringsystem.
 8. A computer program product of predicting low-frequency sensorsignal predictions using a prediction model, the computer programproduct comprising: one or more computer readable storage medium, andprogram instructions stored on the one or more computer readable storagemedia, the program instructions comprising: program instructions toreceive a historical dataset of sensor signal data relating to anenvironment of a sensor monitoring system, wherein the historicaldataset includes a first sensor signal data, a second sensor signaldata, and input variables relating to the sensor monitoring system;program instructions to generate sensor signal responses relating to thefirst sensor signal data by applying a Gaussian process regression modelto the historical dataset and sensor parameters; program instructions togenerate a hierarchical Gaussian process model that jointly considersmulti-dimensional covariance structures among the input variables, thefirst sensor signal data, and the second sensor signal data; and programinstructions to predict, by the hierarchical Gaussian process model,signal values relating to the second sensor signal data at time periodswhere the second sensor signal data was not monitored at using thesensor signal responses.
 9. The computer program product of claim 8,further comprising: program instructions to generate a surrogateprediction model for the sensor monitoring system to provide a frameworkfor transitioning the sensor monitoring system to a differentenvironment; program instructions to train the surrogate predictionmodel using the historical dataset and a second historical dataset,wherein the second historical dataset includes other first sensor signaldata and other second sensor signal data relating to a secondapplication; program instructions to quantify an uncertainty relating tothe sensor monitoring system in the different environment using thesurrogate prediction model and unspecified parameters of sensors in thesensor monitoring system relating to the different environment; programinstructions to compute parameter settings of unknown parametersrelating to the different environment by applying a kriging technique toknown parameters of the different environment; and program instructionsto provide the surrogate prediction model with the computed parametersettings as the framework for the sensor monitoring system in thedifferent environment.
 10. The computer program product of claim 8,further comprising: program instructions to determine prediction errorsrelating to the sensor parameters by performing a cross-validation ofthe sensor signal responses with the signal values to provide a feedbackcontrol to the sensor monitoring system; program instructions to plotthe prediction errors with all sensor parameters provided by the sensormonitoring system; program instructions to determine the predictionerrors exceeding a predetermined threshold; and program instructions toadjust a reporting frequency for sensors in the sensor monitoring systembased on the prediction errors exceeding the predetermined threshold.11. The computer program product of claim 8, wherein the programinstructions to generate the hierarchical Gaussian process modelcomprise: program instructions to sort the first sensor signal data andthe second sensor signal data in a descending order based on theirmonitoring frequencies to portray their hierarchical structure; programinstructions to determine a covariance among the input variables and ahighest time frequency signal data at a first level of the hierarchicalstructure; program instructions to produce a second level of thehierarchical structure using the covariance of the highest timefrequency sensor signal data and the covariance of a second highest timefrequency sensor signal data; and program instructions to applyadditional levels to the hierarchical structure until covariances of twolowest time frequency sensor signal data is added to a level.
 12. Thecomputer program product of claim 8, wherein the hierarchical Gaussianprocess model predicts sensor signal data relating to the second sensorsignal data based on a correlation between the first sensor signal dataand the second sensor signal data.
 13. The computer program product ofclaim 8, wherein the hierarchical Gaussian process model jointlyconsiders multi-dimensional covariance structures among the inputvariables, the first sensor signal data, and the second sensor signaldata.
 14. The computer program product of claim 8 wherein a posteriorprediction of the hierarchical Gaussian process model imputes missingsensor data in the sensor monitoring system.
 15. A system for predictinglow-frequency sensor signal predictions using a prediction model, thesystem comprising: a memory; a processor; local data storage havingstored thereon computer executable code; a historical dataset of sensorsignal data relating to an environment of a sensor monitoring system,wherein the historical dataset includes a first sensor signal data, asecond sensor signal data, and input variables relating to the sensormonitoring system; a Gaussian process regression model configured togenerate sensor signal responses relating to the first sensor signaldata; and a hierarchical Gaussian process model configured to predictsignal values relating to the second sensor signal data at time periodswhere the second sensor signal data was not monitored at using thesensor signal responses, wherein the hierarchical Gaussian process modeljointly considers multi-dimensional covariance structures among theinput variables, the first sensor signal data, and the second sensorsignal data.
 16. The system of claim 15, further comprising: a feedbackcontrol configured to perform a cross-validation of the sensor signalresponses with the signal values to determine a prediction errorrelating to sensor parameters; and wherein the feedback control isfurther configured to adjust a reporting frequency for sensors based onthe prediction error.
 17. The system of claim 15, further comprising: amodel updating framework configured to update the Gaussian processregression model and the hierarchical Gaussian process model applied toa second application for the sensor monitoring system based on anuncertainty computed by a surrogate prediction model.
 18. The system ofclaim 15, wherein the hierarchical Gaussian process model predictssensor signal data relating to the second sensor signal data based on acorrelation between the first sensor signal data and the second sensorsignal data.
 19. The system of claim 15, wherein the hierarchicalGaussian process model jointly considers multi-dimensional covariancestructures among the input variables, the first sensor signal data, andthe second sensor signal data.
 20. The system of claim 15, wherein aposterior prediction of the hierarchical Gaussian process model imputesmissing sensor data in the sensor monitoring system.